The problem here is that the matching is being performed against a quoted string rather than a named variable - From the documentation for pos:

pos SCALAR
pos     Returns the offset of where the last "m//g" search
        left off for the variable in question ($_ is used
        when the variable is not specified).  May be modi-
        fied to change that offset.  Such modification
        will also influence the "\G" zero-width assertion
        in regular expressions.  See perlre and perlop.
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If your code was modified thus, matching against a variable rather than a quoted string, it would execute without error:

my $var = 'the quick brown fox';

while( $var =~ m/(.)/g ) {
    print "Matched '$1' \@ pos:", pos($var), "\n";
}
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perl -le 'print+unpack("N",pack("B32","00000000000000000000000111111001"))'